Use the definition of the derivative to evaluate the following...

Question

Use the definition of the derivative to evaluate the following limits.

$\lim_{h \to 0} {\frac{\ln (e^{8} + h) - 8}{h}}_{}$

Accepted Answer

Welcome back everyone. In this problem, we want to evaluate the limit of the natural log of E to the fifth plus H minus five all divided by H as it approaches zero. With the help of the definition of the derivative. For our answer choices, A says the answer is 1/5 B, one C one divided by E to the fifth and de to the fifth. Now to evaluate this limit, the definition of the derivative is going to be very helpful. OK. So what's this definition that we're talking about here? Well, recall OK that in our limit, we're trying to find a natural log of an expression minus a constant divided by our variable and for the natural log of uh an expression, OK. If we are differentiating the natural log of X, OK, then the result is going to be the reciprocal of X or one divided by X. OK. So now if we look at our, the term for our natural log, OK, then the not the derivative sorry of the natural log of E to the fifth. Using our definition here would be equal to the reciprocal of E to the fifth or one divided by E to the fifth. Therefore, the limit OK. Of the natural log of E to the fifth plus H minus five all divided by H as H approaches zero equals one divided by E to the fifth. Therefore, C is the correct answer. Thanks a lot for watching everyone. I hope this video helped.

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Use the definition of the derivative to evaluate the following limits.
$\lim_{h \to 0} {\frac{\ln (e^{8} + h) - 8}{h}}_{}$